Question on MAC Forgery

Question

In MAC forgery, is it assumed that the attacker has the secret key or not? I find this in Wikipedia but still not clear.

Existential forgery is the creation (by an adversary) of at least one message/signature pair, ( m , σ ), where σ was not produced by the legitimate signer. The adversary need not have any control over m, m need not have any particular meaning; the message content is irrelevant — as long as the pair, ( m , σ ), is valid, the adversary has succeeded in constructing an existential forgery.

Answer 1

The key part of the entry is "...where σ was not produced by the legitimate signer.". The legitimate signer is the key holder. We assume the adversary does not possess the key, but would like to have a way to produce MAC tags anyways.

Answer 2

Definition 9.7 in chapter 9 of the the online Handbook of Applied Cryptography defines MACs, and says (my boldface):

Furthermore, given a description of the function family $h$, for every fixed allowable value of $k$ (unknown to an adversary), the following property holds: [...definition of computation-resistance property...]

The following text just after the definition also doesn't make sense unless the key $k$ is secret:

If computation-resistance does not hold, a MAC algorithm is subject to MAC forgery. While computation-resistance implies the property of key non-recovery (it must be computationally infeasible to recover $k$, given one or more text-MAC pairs $(x_i, h_k(x_i))$ for that $k$), key non-recovery does not imply computation-resistance (a key need not always actually be recovered to forge new MACs).

The Handbook is a bit dated, but it's great for these basic definitions.